SUBROUTINE SB04RX( RC, UL, M, A, LDA, LAMBD1, LAMBD2, LAMBD3,
$ LAMBD4, D, TOL, IWORK, DWORK, LDDWOR, INFO )
C
C PURPOSE
C
C To solve a system of equations in quasi-Hessenberg form
C (Hessenberg form plus two consecutive offdiagonals) with two
C right-hand sides.
C
C ARGUMENTS
C
C Mode Parameters
C
C RC CHARACTER*1
C Indicates processing by columns or rows, as follows:
C = 'R': Row transformations are applied;
C = 'C': Column transformations are applied.
C
C UL CHARACTER*1
C Indicates whether A is upper or lower Hessenberg matrix,
C as follows:
C = 'U': A is upper Hessenberg;
C = 'L': A is lower Hessenberg.
C
C Input/Output Parameters
C
C M (input) INTEGER
C The order of the matrix A. M >= 0.
C
C A (input) DOUBLE PRECISION array, dimension (LDA,M)
C The leading M-by-M part of this array must contain a
C matrix A in Hessenberg form.
C
C LDA INTEGER
C The leading dimension of array A. LDA >= MAX(1,M).
C
C LAMBD1, (input) DOUBLE PRECISION
C LAMBD2, These variables must contain the 2-by-2 block to be
C LAMBD3, multiplied to the elements of A.
C LAMBD4
C
C D (input/output) DOUBLE PRECISION array, dimension (2*M)
C On entry, this array must contain the two right-hand
C side vectors of the quasi-Hessenberg system, stored
C row-wise.
C On exit, if INFO = 0, this array contains the two solution
C vectors of the quasi-Hessenberg system, stored row-wise.
C
C Tolerances
C
C TOL DOUBLE PRECISION
C The tolerance to be used to test for near singularity of
C the triangular factor R of the quasi-Hessenberg matrix.
C A matrix whose estimated condition number is less
C than 1/TOL is considered to be nonsingular.
C
C Workspace
C
C IWORK INTEGER array, dimension (2*M)
C
C DWORK DOUBLE PRECISION array, dimension (LDDWOR,2*M+3)
C The leading 2*M-by-2*M part of this array is used for
C computing the triangular factor of the QR decomposition
C of the quasi-Hessenberg matrix. The remaining 6*M elements
C are used as workspace for the computation of the
C reciprocal condition estimate.
C
C LDDWOR INTEGER
C The leading dimension of array DWORK.
C LDDWOR >= MAX(1,2*M).
C
C Error Indicator
C
C INFO INTEGER
C = 0: successful exit;
C = 1: if the quasi-Hessenberg matrix is (numerically)
C singular. That is, its estimated reciprocal
C condition number is less than or equal to TOL.
C
C NUMERICAL ASPECTS
C
C None.
C
C CONTRIBUTORS
C
C D. Sima, University of Bucharest, May 2000.
C
C REVISIONS
C
C -
C
C Note that RC, UL, M, LDA, and LDDWOR must be such that the value
C of the LOGICAL variable OK in the following statement is true.
C
C OK = ( ( UL.EQ.'U' ) .OR. ( UL.EQ.'u' ) .OR.
C ( UL.EQ.'L' ) .OR. ( UL.EQ.'l' ) )
C .AND.
C ( ( RC.EQ.'R' ) .OR. ( RC.EQ.'r' ) .OR.
C ( RC.EQ.'C' ) .OR. ( RC.EQ.'c' ) )
C .AND.
C ( M.GE.0 )
C .AND.
C ( LDA.GE.MAX( 1, M ) )
C .AND.
C ( LDDWOR.GE.MAX( 1, 2*M ) )
C
C These conditions are not checked by the routine.
C
C KEYWORDS
C
C Hessenberg form, orthogonal transformation, real Schur form,
C Sylvester equation.
C
C ******************************************************************
C
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0 )
C .. Scalar Arguments ..
CHARACTER RC, UL
INTEGER INFO, LDA, LDDWOR, M
DOUBLE PRECISION LAMBD1, LAMBD2, LAMBD3, LAMBD4, TOL
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), D(*), DWORK(LDDWOR,*)
C .. Local Scalars ..
CHARACTER TRANS
INTEGER J, J1, J2, M2, MJ, ML
DOUBLE PRECISION C, R, RCOND, S
C .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
C .. External Subroutines ..
EXTERNAL DCOPY, DLARTG, DLASET, DROT, DSCAL, DTRCON,
$ DTRSV
C .. Intrinsic Functions ..
INTRINSIC MAX, MIN, MOD
C .. Executable Statements ..
C
INFO = 0
C
C For speed, no tests on the input scalar arguments are made.
C Quick return if possible.
C
IF ( M.EQ.0 )
$ RETURN
C
M2 = M*2
IF ( LSAME( UL, 'U' ) ) THEN
C
DO 20 J = 1, M
J2 = J*2
ML = MIN( M, J + 1 )
CALL DLASET( 'Full', M2, 2, ZERO, ZERO, DWORK(1,J2-1),
$ LDDWOR )
CALL DCOPY( ML, A(1,J), 1, DWORK(1,J2-1), 2 )
CALL DSCAL( ML, LAMBD1, DWORK(1,J2-1), 2 )
CALL DCOPY( ML, A(1,J), 1, DWORK(2,J2-1), 2 )
CALL DSCAL( ML, LAMBD3, DWORK(2,J2-1), 2 )
CALL DCOPY( ML, A(1,J), 1, DWORK(1,J2), 2 )
CALL DSCAL( ML, LAMBD2, DWORK(1,J2), 2 )
CALL DCOPY( ML, A(1,J), 1, DWORK(2,J2), 2 )
CALL DSCAL( ML, LAMBD4, DWORK(2,J2), 2 )
C
DWORK(J2-1,J2-1) = DWORK(J2-1,J2-1) + ONE
DWORK(J2,J2) = DWORK(J2,J2) + ONE
20 CONTINUE
C
IF ( LSAME( RC, 'R' ) ) THEN
TRANS = 'N'
C
C A is an upper Hessenberg matrix, row transformations.
C
DO 40 J = 1, M2 - 1
MJ = M2 - J
IF ( MOD(J,2).EQ.1 .AND. J.LT.M2-2 ) THEN
IF ( DWORK(J+3,J).NE.ZERO ) THEN
CALL DLARTG( DWORK(J+2,J), DWORK(J+3,J), C, S, R )
DWORK(J+2,J) = R
DWORK(J+3,J) = ZERO
CALL DROT( MJ, DWORK(J+2,J+1), LDDWOR,
$ DWORK(J+3,J+1), LDDWOR, C, S )
CALL DROT( 1, D(J+2), 1, D(J+3), 1, C, S )
END IF
END IF
IF ( J.LT.M2-1 ) THEN
IF ( DWORK(J+2,J).NE.ZERO ) THEN
CALL DLARTG( DWORK(J+1,J), DWORK(J+2,J), C, S, R )
DWORK(J+1,J) = R
DWORK(J+2,J) = ZERO
CALL DROT( MJ, DWORK(J+1,J+1), LDDWOR,
$ DWORK(J+2,J+1), LDDWOR, C, S )
CALL DROT( 1, D(J+1), 1, D(J+2), 1, C, S )
END IF
END IF
IF ( DWORK(J+1,J).NE.ZERO ) THEN
CALL DLARTG( DWORK(J,J), DWORK(J+1,J), C, S, R )
DWORK(J,J) = R
DWORK(J+1,J) = ZERO
CALL DROT( MJ, DWORK(J,J+1), LDDWOR, DWORK(J+1,J+1),
$ LDDWOR, C, S )
CALL DROT( 1, D(J), 1, D(J+1), 1, C, S )
END IF
40 CONTINUE
C
ELSE
TRANS = 'T'
C
C A is an upper Hessenberg matrix, column transformations.
C
DO 60 J = 1, M2 - 1
MJ = M2 - J
IF ( MOD(J,2).EQ.1 .AND. J.LT.M2-2 ) THEN
IF ( DWORK(MJ+1,MJ-2).NE.ZERO ) THEN
CALL DLARTG( DWORK(MJ+1,MJ-1), DWORK(MJ+1,MJ-2), C,
$ S, R )
DWORK(MJ+1,MJ-1) = R
DWORK(MJ+1,MJ-2) = ZERO
CALL DROT( MJ, DWORK(1,MJ-1), 1, DWORK(1,MJ-2), 1,
$ C, S )
CALL DROT( 1, D(MJ-1), 1, D(MJ-2), 1, C, S )
END IF
END IF
IF ( J.LT.M2-1 ) THEN
IF ( DWORK(MJ+1,MJ-1).NE.ZERO ) THEN
CALL DLARTG( DWORK(MJ+1,MJ), DWORK(MJ+1,MJ-1), C,
$ S, R )
DWORK(MJ+1,MJ) = R
DWORK(MJ+1,MJ-1) = ZERO
CALL DROT( MJ, DWORK(1,MJ), 1, DWORK(1,MJ-1), 1, C,
$ S )
CALL DROT( 1, D(MJ), 1, D(MJ-1), 1, C, S )
END IF
END IF
IF ( DWORK(MJ+1,MJ).NE.ZERO ) THEN
CALL DLARTG( DWORK(MJ+1,MJ+1), DWORK(MJ+1,MJ), C, S,
$ R )
DWORK(MJ+1,MJ+1) = R
DWORK(MJ+1,MJ) = ZERO
CALL DROT( MJ, DWORK(1,MJ+1), 1, DWORK(1,MJ), 1, C,
$ S )
CALL DROT( 1, D(MJ+1), 1, D(MJ), 1, C, S )
END IF
60 CONTINUE
C
END IF
ELSE
C
DO 80 J = 1, M
J2 = J*2
J1 = MAX( J - 1, 1 )
ML = MIN( M - J + 2, M )
CALL DLASET( 'Full', M2, 2, ZERO, ZERO, DWORK(1,J2-1),
$ LDDWOR )
CALL DCOPY( ML, A(J1,J), 1, DWORK(J1*2-1,J2-1), 2 )
CALL DSCAL( ML, LAMBD1, DWORK(J1*2-1,J2-1), 2 )
CALL DCOPY( ML, A(J1,J), 1, DWORK(J1*2,J2-1), 2 )
CALL DSCAL( ML, LAMBD3, DWORK(J1*2,J2-1), 2 )
CALL DCOPY( ML, A(J1,J), 1, DWORK(J1*2-1,J2), 2 )
CALL DSCAL( ML, LAMBD2, DWORK(J1*2-1,J2), 2 )
CALL DCOPY( ML, A(J1,J), 1, DWORK(J1*2,J2), 2 )
CALL DSCAL( ML, LAMBD4, DWORK(J1*2,J2), 2 )
C
DWORK(J2-1,J2-1) = DWORK(J2-1,J2-1) + ONE
DWORK(J2,J2) = DWORK(J2,J2) + ONE
80 CONTINUE
C
IF ( LSAME( RC, 'R' ) ) THEN
TRANS = 'N'
C
C A is a lower Hessenberg matrix, row transformations.
C
DO 100 J = 1, M2 - 1
MJ = M2 - J
IF ( MOD(J,2).EQ.1 .AND. J.LT.M2-2 ) THEN
IF ( DWORK(MJ-2,MJ+1).NE.ZERO ) THEN
CALL DLARTG( DWORK(MJ-1,MJ+1), DWORK(MJ-2,MJ+1), C,
$ S, R )
DWORK(MJ-1,MJ+1) = R
DWORK(MJ-2,MJ+1) = ZERO
CALL DROT( MJ, DWORK(MJ-1,1), LDDWOR,
$ DWORK(MJ-2,1), LDDWOR, C, S )
CALL DROT( 1, D(MJ-1), 1, D(MJ-2), 1, C, S )
END IF
END IF
IF ( J.LT.M2-1 ) THEN
IF ( DWORK(MJ-1,MJ+1).NE.ZERO ) THEN
CALL DLARTG( DWORK(MJ,MJ+1), DWORK(MJ-1,MJ+1), C,
$ S, R )
DWORK(MJ,MJ+1) = R
DWORK(MJ-1,MJ+1) = ZERO
CALL DROT( MJ, DWORK(MJ,1), LDDWOR, DWORK(MJ-1,1),
$ LDDWOR, C, S )
CALL DROT( 1, D(MJ), 1, D(MJ-1), 1, C, S )
END IF
END IF
IF ( DWORK(MJ,MJ+1).NE.ZERO ) THEN
CALL DLARTG( DWORK(MJ+1,MJ+1), DWORK(MJ,MJ+1), C, S,
$ R )
DWORK(MJ+1,MJ+1) = R
DWORK(MJ,MJ+1) = ZERO
CALL DROT( MJ, DWORK(MJ+1,1), LDDWOR, DWORK(MJ,1),
$ LDDWOR, C, S)
CALL DROT( 1, D(MJ+1), 1, D(MJ), 1, C, S )
END IF
100 CONTINUE
C
ELSE
TRANS = 'T'
C
C A is a lower Hessenberg matrix, column transformations.
C
DO 120 J = 1, M2 - 1
MJ = M2 - J
IF ( MOD(J,2).EQ.1 .AND. J.LT.M2-2 ) THEN
IF ( DWORK(J,J+3).NE.ZERO ) THEN
CALL DLARTG( DWORK(J,J+2), DWORK(J,J+3), C, S, R )
DWORK(J,J+2) = R
DWORK(J,J+3) = ZERO
CALL DROT( MJ, DWORK(J+1,J+2), 1, DWORK(J+1,J+3),
$ 1, C, S )
CALL DROT( 1, D(J+2), 1, D(J+3), 1, C, S )
END IF
END IF
IF ( J.LT.M2-1 ) THEN
IF ( DWORK(J,J+2).NE.ZERO ) THEN
CALL DLARTG( DWORK(J,J+1), DWORK(J,J+2), C, S, R )
DWORK(J,J+1) = R
DWORK(J,J+2) = ZERO
CALL DROT( MJ, DWORK(J+1,J+1), 1, DWORK(J+1,J+2),
$ 1, C, S )
CALL DROT( 1, D(J+1), 1, D(J+2), 1, C, S )
END IF
END IF
IF ( DWORK(J,J+1).NE.ZERO ) THEN
CALL DLARTG( DWORK(J,J), DWORK(J,J+1), C, S, R )
DWORK(J,J) = R
DWORK(J,J+1) = ZERO
CALL DROT( MJ, DWORK(J+1,J), 1, DWORK(J+1,J+1), 1, C,
$ S )
CALL DROT( 1, D(J), 1, D(J+1), 1, C, S )
END IF
120 CONTINUE
C
END IF
END IF
C
CALL DTRCON( '1-norm', UL, 'Non-unit', M2, DWORK, LDDWOR, RCOND,
$ DWORK(1,M2+1), IWORK, INFO )
IF ( RCOND.LE.TOL ) THEN
INFO = 1
ELSE
CALL DTRSV( UL, TRANS, 'Non-unit', M2, DWORK, LDDWOR, D, 1 )
END IF
C
RETURN
C *** Last line of SB04RX ***
END